Theorem sucexeloni 7756

Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7757 does not require ax-un 7682. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)

Assertion

Ref	Expression
sucexeloni	⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni

Step	Hyp	Ref	Expression
1		eloni 6327	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsuci 7755	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		elex 3451	. 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
5		elong 6325	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
6	5	biimparc 479	. 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
7	3, 4, 6	syl2an 597	1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3430  Ord word 6316  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  onsuc  7757  1on  8410  2on  8411